Chapter 4, Fluid Kinematics

Chapter 4, Fluid Kinematics

To study the fluid motion without the consideration of the forces necessary to produce the motion

1. Velocity Field

V = u(x,y,z,t)i + vj + wk

|V| =(u2 + v2 + w2)1/2

i) Description

Langrangian flow description: following the individual fluid particles (Closed System)

Eulerian flow description: obtaining information about the flow in terms of what happen at fixed points in space as the flow past these points (Control Volume)

ii) Fluid flow is generally three-dimensional

V = u(x,y,z,t)i + vj + wk

It may be simplified to

one-dimensional, V = u(x,t)i Pipe flow

two-dimensional, V = u(x,t)i + v(y,t)j around a cylinder

iii) Unsteady and Turbulent

Assume steady ∂V/∂t = 0

laminar Re < 2300 for pipe flows

2. The Acceleration Field

Velocity

V = u(x,y,z,t)i + vj + wk

Acceleration

a = dV/dt = ∂V/∂t + (∂V/∂x)(dx/dt) + (∂V/∂y)(dy/dt)

+ (∂V/∂z)(dz/dt)

Since u = dx/dt, v = dy/dt, w = dz/dt

i) Material derivative

a = DV/Dt

= ∂V/∂t + u(∂V/∂x) + v(∂V/∂y) + w(∂V/∂z)

= ∂V/∂t + V∙V

where the vector operator,

del =  = i∂/∂x + j∂/∂y + k∂/∂z

3. Control Volume and System Representation

System: consider all mass as the system (Lagrangian)

Control Volume: consider a space through which mass may flow in and out (Eulerian)

4. Reynolds Transport Theorem

At time t, Sys(t) = CV(t)

At time t = δt,

Sys (t+δt) = [(CV -I) + II](t+δt)

Let B = the extensive property

B = mb = ∫ ρbdV

Bsys(t+δt) = BCV(t+δt) - BI(t+δt) +BII(t+δt)

δBsys(t+δt)/δt = [Bsys(t+δt) - Bsys(t)]/δt

= [BCV(t+δt) - BCV(t)]/δt - BI(t+δt)/δt + BII(t+δt)/δt

As δt  0, Reynolds Transport Theorem

DBsys/Dt = ∂BCV/∂t - Bin + Bout

= (∂/∂t)∫ bρdV + ∫ bρV∙ndA

CV CS

In one-dimensional flow

DBsys/Dt = ∂BCV/∂t + [(ρAVb)out - (ρAVb)in]CS